I am a musician/producer with a fairly discerning ear, and have been following this discussion for many months with great interest. Basically, there are two different problems with the current stock Apple Music app that are somewhat related, and are causing all the complaints documented here. To summarize:
Firstly, the stock Music app now uses some newly re-written code introduced with iOS5 to reproduce all music that results in a more brittle, harsh and much less pleasing sound overall on pretty much every track when compared to the "classic" sound of the pre-iOS 5 music app. Part of the distortion problem everyone is complaining about here is a result of this "new" code.
Secondly, it introduces even more distortion on tracks that have their volume boosted in the iTunes "Options" tab in the "Get Info" screen of each song. In addition, there is a wierd glitch that causes it to "remember" a boosted volume setting and apply it to following tracks in a playlist, even if those next tracks are not boosted. I have over a thousand songs on my iPhone, and only some have boosted Volumes in their Options. (I only boosted the tracks that I felt were too soft to begin with, but left other tracks un-changed.) When I'm listening to any playlist or in shuffle mode, the tracks that follow a "boosted" track STAY boosted even if they were not supposed to be! That's most likely another source of the "bass distortion" everyone is complaining about. When that happens, I discovered that manually going back to the previous track and then immediately manually skipping foward to that same next track seems to "clear" the error ... and the track that was distorted suddenly plays correctly without the volume boost and related distortion.
The solution offered here to use the "Music+" app, which has recently been re-named "Lagu" (terrible name, by the way!) is a perfect one that I highly recommend any serious music enthusiast or audiophile should try. It fixes both of the above mentioned problems. The app is free, and includes several features that the stock Music app does not have -- like the ability to create instant "Queque" lists on the fly, and to retrieve lyrics. I've done many side-by-side listening tests comparing playbacks of the same exact tracks (with the Options Volume at 0%) played through "Music" vs. "Lagu" and the results are quite obvious and startling. "Lagu" sounds GREAT -- smooth and rich, just like the sound of the old iPods we used to love and cherish. No harshness, and no bass distortion. You can now even use a built-in equalizer function with a long list of pre-sets from within the app itself. One thing it does NOT do is pay attention to the Volume adjustments made in iTunes -- but since that was one source of the distortion to begin with ... that's probably a good thing!
One last "perk" of Lagu (for me, anyway) is that it makes use of the older style visual interface on the iPhone screen. By that, I mean the thinner Blue & White sliders and indicators instead of the bigger, fatter silvery sliders and buttons and Orange indicators on the newest "Music" app. I find those older controls much more attractive to look at and precise to use -- but that's just my opinion!